Interior and Exterior Angles of Polygons Practice - PDF

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Okay, here is the converted text from the images into a structured markdown format. I have focused on retaining information, mathematical formulas, and table structures while aiming for clarity and readability. ### Page 1 Name: Class: Topic: Date: **Main Ideas/Questions | Notes** ------------ | ------------- **Interior Angles Sum** | The sum of the degrees in any polygon can be determined by the number of triangles that can be drawn within the polygon. Complete the table below and look for a pattern to find the sum of the degrees in any polgon. **Table of Interior Angles** | Polygon | Picture | # of Sides | # of Triangles | Sum of Interior Angles | | ------------- | ------------------------------------------- | ---------- | -------------- | ---------------------- | | Triangle | Triangle Image | 3 | 1 | 180° | | Quadrilateral | Quadrilateral Image | 4 | 2 | 360° | | Pentagon | Pentagon Image | 5 | 3 | 540° | | Hexagon | Hexagon Image | 6 | 4 | 720° | | Heptagon | Image marked with an "X" | 7 | 5 | 900° | | Octagon | Image marked with an "X" | 8 | 6 | 1080° | | Nonagon | Image marked with an "X" | 9 | 7 | 1260° | | Decagon | Image marked with an "X" | 10 | 8 | 1440° | **Sum of Interior Angles Formula!** | If n represents the number of sides of a polygon, then the sum of the interior angles (S) can be found using the formula: | $S= (n-2) \cdot 180$ **Interior Angle of a Regular Polygon** | A regular polygon is one is which all sides are equal; therefore all angles are equal! To find the measure of an interior angle in a regular polygon, use: | $\frac{S}{n}$ | (Sum of int $\angle$'s $\div$ # of sides) Gina Wilson (All Things Algebra), 2014 ### Page 2 **Interior Angles Practice!** 1. Find the sum of the interior angles of a 15 gon. $S = (15-2) \cdot 180 = 2340°$ 2. Find the sum of the interior angles of a 21-gon. $S = (21-2) \cdot 180 = 3420°$ 3. What is the measure of each interior angle of a regular pentagon? $S = (5 - 2) \cdot 180 = 540$ $540/5 = 108°$ 4. What is the measure of each interior angle of a regular 18-gon? $S = (18-2) \cdot 180 = 2880$ $2880/18 = 160°$ **Exterior Angles Sum** Exterior angles are always supplementary to their adjacent interior angle. On the polygons below, find the measure of each exterior angle along with the sum of all exterior angles. *Triangle:* Image of triangle with exterior and interior angles labeled, Sum of Exterior Angles: $360°$ *Quadrilateral:* Image of quadrilateral with exterior and interior angles labeled, Sum of Exterior Angles: $360°$ *Pentagon:* Image of pentagon with exterior and interior angles labeled, Sum of Exterior Angles: $360°$ *Hexagon:* Image of hexagon with exterior and interior angles labeled, Sum of Exterior Angles: $360°$ Conclusion: The sum of the exterior angle of any polygon is $360°$ **Exterior Angles Practice!** 5. What is the measure of each exterior angle of a regular hexagon? $\frac{360}{6} = 60°$ 6. What is the measure of each exterior angle of a regular 24-gon? $\frac{360}{24} = 15°$ 7. If the exterior angle of a regular polygon measures $12°$, how many sides does the polygon have? $\frac{360}{x} = 12$ $12x = 360$ $x = 30$ 8. If the exterior angle of a regular polygon measures $40°$, how many sides does the polygon have? $\frac{360}{x} = 40$ $40x = 360$ $x = 9$ Gina Wilson (All Things Algebra), 2014 ### Page 3 **PRACTICE: Interior & Exterior Angles of Polygons** **Formulas Review** | Formula | | ----------- | | Sum of Interior Angles of a Polygon: $S = (n-2) \cdot 180$ | | Interior Angle of a Regular Polygon: $\frac{S}{n}$ | | Sum of Exterior Angles of a Polygon: $360°$ | | Exterior Angle of a Regular Polygon: $\frac{360}{n}$ | | Number of Sides of a Regular Polygon: $\frac{360}{\text{ext } \angle}$ | **Practice Questions** 1. What is the sum of the measures of the interior angles of a pentagon? $S = (5-2) \cdot 180 = 540°$ 2. What is the sum of the measures of the interior angles of a 27-gon? $S = (27-2) \cdot 180 = 4500°$ 3. What is the measure of each interior angle of a regular octagon? $S = (8-2) \cdot 180 = 1080$ $1080/8 = 135°$ 4. What is the measure of *each* interior angle of a regular 20-gon? $S=(20-2) \cdot 180 = 3240$ $3240/20=162°$ 5. Five angles of a hexagon measure $119°$, $129°$, $104°$, $139°$, and $95°$. What is the measure of the sixth angle? $S = (6-2) \cdot 180 = 720°$ $720-586 = 134°$ 6. The sum of the interior angles of a polygon is $1620°$. How many sides does the polygon have? $1620 = (n-2) \cdot 180$ $9 = n-2$ $11 = n$ 7. The sum of the interior angles of a polygon is $3960°$. How many sides does the polygon have? $3960 = (n-2) \cdot 180$ $22 = n-2$ $24 = n$ 8. What is the sum of the measures of the exterior angles of a nonagon? $360°$ 9. What is the measure of each exterior angle of a 20-gon? $\frac{360}{20} = 18°$ Gina Wilson (All Things Algebra), 2014 ### Page 4 10. If the exterior angle of a regular polygon measures $9°$, how many sides does the polygon have? $\frac{360}{n} = 9$ $9n = 360$ $n = 40$ 11. If the interior angle of a regular polygon measures $108°$, how many sides does the polygon have? ext $\angle = 12$ $\frac{360}{n} = 72$ $72n=360$ $n = 5$ 12. Find the value of x. Image of a quadrilateral with angles labeled. $360-244=116°$ 13. Find the value of x. Image of a quadrilateral with angles labeled. $900-806 = 94°$ 14. Solve for x. Image of a polygon with angles labeled in terms of x $110 + 8x - 1 + 5x + 36 + 116 + 7x + 19 + 6x - 2 = 720$ $26x + 278 = 720$ $26x = 442$ $x = 17$ 15. Find m∠B. Image of a quadrilateral ABCD $8x+7 + 5x+18 + 10x+13 + 14x -11 = 360$ $37x + 27 = 360$ $37x = 333$ $x = 9$ $m\angle B = 8(9) + 7 = 79°$ 16. Solve for x. Image of a polygon with angles labeled $10(10x + 4) = 1440$ $100x + 40 = 1440$ $100x = 1400$ $x = 14$ 17. Solve for x. Image of a polygon with angles labeled in terms of x $5x + 4 + 4x + 9 + 9x -6 + 4x + 1 + 7x + 4 = 360$ $29x + 12 = 360$ $29x = 348$ $x = 12$ Gina Wilson (All Things Algebra), 2014 ### Page 5 Name: Class: Topic: Date: **Main Ideas/Questions | Notes** ------------ | ------------- **Properties of Parallelograms** | 1. Opposite sides are parallel. | | 2. Opposite sides are congruent. | | 3. Opposite angles are congruent. | | 4. Consecutive angles are supplementary. | | 5. Diagonals bisect each other. **Directions:** Each quadrilateral below is a parallelogram. Find the missing measures. 1. Image of parallelogram ABCD with AD=8, AB=15, DC=15 $AD = 8$ $DC = 15$ $m \angle A = 112°$ $m \angle B =68°$ $m \angle C = 112°$ 2. Image of parallelogram JKML with JK=21, KL=29 $JK = 21$ $KL = 29$ $m \angle J = 127°$ $m \angle K = 53°$ $m \angle M = 53°$ 3. Image of parallelogram RSTU with RT=30 $UT = 27$ $ST = 18$ $VS = 7$ $VT = 15$ 4. Image of parallelogram CDEF $m\angle DEC= 63°$ $m \angle CDE= 46°$ $m \angle ECD = 71°$ $m \angle FED= 134°$ $m \angle DFE= 25°$ 5. Given XY = 15, WX = 22, ZX = 52, WT = 23,$m \angle WZY= 72°$, $m \angle WXT = 35°$, and $m \angle ZWT = 59°$. Image of parallelogram WXYZ with diagonal XT $WZ= 15$ $ZY = 22$ $TX=26$ $WY=46$ $m \angle TZY= 35°$ $m\angle WZY= 72°$ $m \angle XYZ= 108°$ $m \angle XYT= 49°$ 6. Given $m \angle GHF= 34°$, $m\angle HJF = 124°$, and $m \angle FKJ = 79°$. Image of parallelogram FGJH with segment FK $m\angle FGJ= 56°$ $m\angle FGH= 124°$ $m\angle JGH= 67°$ $m\angle HFJ= 34°$ $m\angle FGJ= 61°$ $m\angle HKJ= 181°$ $m\angle FHJ= 22°$ $m\angle GJF= 67°$ Gina Wilson (All Things Algebra), 2014 ### Page 6 7. Solve for x. Image of Parallelogram $9x-25= 5x+7$ $4x = 32$ $x=8$ 8. Find YZ. Image of Parallelogram $19x-31= 11x+1$ $8x = 32$ $x\angle B = 4$ $YZ= 19(4)-31= 45$ 9. If TV = 74 and WV = $4x+1$, solve for x. Image of Parallelogram $2(4x +1)=74$ $8x+2=74$ $8x=72$ $x=9$ 10. if NS = $2x+7$ and SQ = $5x-23$, Find NQ. Image of Parallelogram $5x -23=2x+7$ $3x=30$ $x = 10$ $NS= 2(10)+7 = 27$ $NQ= 2(27)= 54$ 11. Find $m\angle B$ Image of Parallelogram $6x -15= 4x +11$ $2x = 26$ $x = 13$ $m\angle C = 4(13) +11 = 63$ $m\angle B = 180 - 63 = 117°$ 12. Find $m\angle R$ Image of Parallelogram $8x+3x+5=180$ $11x -7=180$ $11x \angle B = 187$ $x = 17$ $m\angle P = m\angle R$ $m\angle P = 3 (17) +5 = 56°$ 13. If $m\angle KLH = 134°$, Solve for x. Image of Parallelogram $4x =3$ $4x + 9 +25+134 =180$ $4x=12$ 14. If $m\angle ABC = 1115°$ ,find $m\angle ADB$. Image of Parallelogram $ \angle B = 6x -11+4x+6=115$ $ 10x-5=115$ $10x =3$ mLADB = 6 (12) - 11 = 61° Gina Wilson (All Things Algebra), 2014 ### Page 7 **Proving Parallelograms in the Coordinate Plane** | Method | Description | Instructions | | ------ | ------ | ------ | | Method 1 | Prove both pairs of opposite sides are congruent. | If $\overline{AB} \cong \overline{CD}$ and $\overline{DA} \cong \overline{BC}$, then ABCD is a parallelogram. Use the distance formula.| | Method 2 | Prove both pairs of opposite sides are parallel.| If $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \parallel\overline{CB}$, then ABCD is a parallelogram. Use slope formula. | | Method 3 | Prove one pair of opposite sides are congruent and parallel. | If $\overline{AB} \cong \overline{CD}$ and $\overline{AB} \parallel \overline{CD}$, then ABCD is a parallelogram. Use the distance + slope formulas | Directions: Determine whether the figure is a parallelogram using the distance formula. 1. $A(−7, 4), B(1, 2), C(9, −8), D(1, −6)$\ $AB=\sqrt{(1+7)^{2}+(2-4)^{2}}=\sqrt{64+4}=\sqrt{68}$ $BC=\sqrt{(9-1)^{2}+(-8-2)^{2}}=\sqrt{64+100}=\sqrt{164}$ $CD=\sqrt{(1-9)^{2}+(-6+8)^{2}}=\sqrt{64+4} = \sqrt{68}$ $DA=\sqrt{(1+7)^{2}+(-6-4)^{2}}=\sqrt{64 + =100} = \sqrt{164}$ YES! ABCD is a Parrallelogram 2. $P(−4, 2), Q(6, 4), R(11, −2), S(2, −3)$\ $PQ=\sqrt{(6+4)^{2}+(4-2)^{2}}=\sqrt{100+4}=√104$\ $RS=\sqrt{(2-11)^{2}+(-3+2)^{2}}=\sqrt{81+1}=√82$\ $\overline{PQ} \ncong \overline{RS}$ No! PQRS is not a Parallelogram Gina Wilson (All Things Algebra), 2014 ### Page 8 Directions: Determine whether the figure is a parallelogram using the slope formula. 3. W(−7, −4), X(1, −6), Y(5, −13), Z(−12) $m\overline{WX} = \frac{-6 + 4}{1 + 7} = \frac{-2}{8}=-\frac{1}{4}$ $m\overline{XY} = \frac{-13 + 6}{5 - 1} = \frac{-7}{4}$ $m\overline{WZ} = \frac{-12 + 4}{1 + 7} = \frac{-8}{8}= -1$ NO! WXYZ is not a parallelogram. 4. E(0, 8), F(6, 10), G(2, 0), H(−4, −2) $m\overline{EF} = \frac{10 - 8}{6-0}=\frac{2}{6} =\frac{1}{3}$ $m\overline{GH} = \frac{-2 - 0}{- 4 - 2} = \frac{-2}{- 6} = \frac{1}{3}$ $m\overline{FG} = \frac{0 - 10}{2- 6} = \frac{-10}{- 4} = \frac{ 5}{2}$ $m\overline{EH} = \frac{-2 - 8}{- 4 - 0} = \frac{-10}{- 4} = \frac{ 5}{2}$ YES! EFGH is a parrallelogram Directions: Determine whether the figure is a parallelogram using the distance and slope formulas 5. J(−9, −2), K(−5, 1), L(1, −4), M(−3, −7) $JK=\sqrt{(-5+ 9)^{2}+(1 + 2)^{2}} = \sqrt{16 + 9} = \sqrt{25}= 5$ $LM=\sqrt{(-1+ 3)^{2}+(-1+ 7)^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5$ $m\overline{JK} = \frac{1 + 2}{- 6 + 9}= \frac{3}{4}$ $m\overline{LM} = \frac{5 + 4}{- 3 - 1} = \frac { 5}{4}$ YES! JKLM is a parallelogram 6. S(1, 5), T(10, 7), U(14, 1), V(5, −1) $ST = \sqrt{(10 - 1)^{2}+(7 - 5)^{2}} = \sqrt{81 + 4} = √(\frac{85}$ $UV = \sqrt{(5 - 11)^{2}+(-1 - 1)^{2}} = \sqrt{149 + 4} = {√\frac{33}{}}$ No! STUV is not a parallelogram GIna Wilson (All Things Algebra), 2014